Remarks on mass transportation minimizing expectation of a minimum of affine functions

Abstract: We study the Monge--Kantorovich problem with one-dimensional marginals $\mu$ and $\nu$ and the cost function $c = \min{l_1, \ldots, l_n}$ that equals the minimum of a finite number $n$ of affine functions $l_i$ satisfying certain non-degeneracy assumptions. We prove that the problem is equivalent to a finite-dimensional extremal problem. More precisely, it is shown that the solution is concentrated on the union of $n$ products $I_i \times J_i$, where ${I_i}$ and ${J_i}$ are partitions of the real line into unions of disjoint connected sets. The families of sets ${I_i}$ and ${J_i}$ have the following properties: 1) $c=l_i$ on $I_i \times J_i$, 2) ${I_i}, {J_i}$ is a couple of partitions solving an auxiliary $n$-dimensional extremal problem. The result is partially generalized to the case of more than two marginals.